Let $K$ be a field, $\overline{K}$ be
a separable closure of $K$ with
absolute Galois group
$G_K:=Gal(\overline{K}/K)$, and let
$\ell$ be a prime that is different
from $char(K)$. Let $X$ be a
$K$-scheme that is separated and of
finite type. The $\ell$-adic
cohomology groups with proper support
$H^i_c(X_{\overline{K}},\mathbb{Q}_{\ell})$.

They are finite dimensional
$\mathbb{Q}_{\ell}$ vector spaces and
are zero for $i>2\cdot \text{dim}(X)$
and $G_K$ acts (via monodromy, as the fundamental group) continuously on them,
so that for each $g\in G_K$, the trace
$$\text{Tr}(g,H^\ast_c(X_{\overline{K}},\mathbb{Q}_{\ell}))=\sum(-1)^i\text{Tr}(g,H^i_c(X_{\overline{K}},\mathbb{Q}_{\ell}))$$
is defined. This $\ell$-adic number is
in fact an $\ell$-adic integer.

Write $K$ as the inductive limit of
its $\mathbb{Z}$-sub-algebras of
finite type. A model of $X/K$ over
such a subring $R$ is a scheme
$\mathcal{X}$ that is separated and of
finite type over $S=Spec(R)$ such that
$\mathcal{X}\times_{S} Spec(K)= X$. A
model of $X/K$ is essentially unique
``up to shrinking'': two models
$\mathcal{X}_1/R_1$ and
$\mathcal{X}_2/R_2$ become isomorphic
over some $S=Spec(R)$ with $R\supseteq
> R_1,R_2$ [EGA4,section 8].

Theorem 1 [Serre, 2004] Under the above notations and assumptions, for a
positive integer $n$, the following
conditions are equivalent:

There exists a model $\mathcal{X}/S$ of $X/K$ having the
following property: for all points
$s=Spec(k')\to S$ with value in a
finite field $k'$ of characteristic
different from $\ell$, we have
$$|\mathcal{X}(s)|\equiv 0\text{ }(
> \text{ mod } \ell^n)$$

This result is published in Illusie's note Miscellany on traces in $\ell$-adic cohomology from 2005.

On the other hand, let me recall Saito's approach (1987) to stable reduction of curves :

let $C$ be a curve proper and smooth over $K=Frac(R)$, $R$ a complete DVR with algebraically closed residual field $k(R)$ of characteristic $p>0$.

Theorem 2 [Stable reduction] Let $C$ be a geometrically irreducible proper smooth curve over $K$ of genus $g\geq 2$. Then there exists a finite separable extension $K'/K$ such that $C_{K'} = C\times_K K'$ admits a stable model over the integral closure $R'$ of $R$ in $K'$. I.e., $C_{K}$ is the generic fibre of some $C_m/R$ which is proper, flat, separated of finite type of relative dimension 1 whose geometric fibres are stable curves (i.e., they are reduced, connected, with only ordinary double points as singularities, and each connected component intersects the others in at least 3 points).

Saito's proof essentially shows that whether a curve admits a stable model (i.e., $K'=K$) is encoded by the representation $H^1(C_{\overline{K}},\mathbb{Q}_{\ell})$ of $Gal(\overline{K}/K)$ :

Theorem [Saito] Let $C$ be a geometrically irreducible proper smooth curve over $K$ of genus $g\geq 2$. The following conditions are equivalent :

$C$ admits a stable model

the action of inertia $I_K$ on $H^1(C_{\overline{K}},\mathbb{Q}_{\ell})$ is unipotent.

I refer you to Abbes' article in Courbes semi-stables et groupe fondamental en géométrie algébrique, Birkhäuser (1998) for a discussion of Saito's proof.

Question Do you know of other theorems of a similar flavor ? Are there general remarks on why such results might not be surprising ?

Answers I'm looking for would : state other theorems of a similar flavor, or explain why such theorems shouldn't be surprising (i.e., by heuristic explainations of how étale cohomology naturally detects such aspects of geometry).

This is not at all an answer to your question, but a friend of mine suggested a possible model-theoretic proof of the infinitude of Mersenne primes some time last year. It seemed to reduce the problem to a harder model theory problem (hence the fact that this proof was nver finished.) It seems to be a similar sort of thing: there's a nice model theoretic way of looking at a problem, and a lot of the work has already been done by model theorists somewhere.
–
Cory KnappApr 29 '10 at 16:15

1

Could you be more specific about what is confusing you? Is your question really just "cohomology is powerful! What's with that?" Also, you're using the term "model theory" in a very misleading way. There is a branch of mathematics called "model theory" that has nothing to do with this post.
–
Ben Webster♦Apr 29 '10 at 16:17

1

I think the sense in which you are using the word "model" is unrelated to what "model-theory" usually means... In any case, the result of Saito goes back to the Neron-Ogg-Shafarevich criterion for good reduction of abelian varieties and Grothendieck's criterion for semi-stable reduction and is perhaps not really surprising once one takes those results into consideration.
–
ulrichApr 29 '10 at 16:24

OK, 'model theory' dropped. In any case it's clear what is meant by models in both cases.
–
xurosApr 29 '10 at 16:28